Page 169 - The Mechatronics Handbook

0066-frame-C09 Page 43 Friday, January 18, 2002 11:01 AM

A 3-port I element can be used to represent the rotational kinetic energy for the case of rotation about
a fixed point (no translation). The constitutive relation is simply Eq. (9.25). The kinetic energy is then

T = 1
--ω h⋅
2
h
where is the angular momentum with an inertia tensor defined about the fixed point. If the axes are
aligned with principal axes, then

--I y ω y +
T = 1 --I x ω x + 1 2 1 --I z ω z 2
2
2 2 2
The total kinetic energy for a rigid body that can translate and rotate, with angular momentum defined
with reference to the center of gravity, is given by

T = 1 --mV G + 1 ⋅
2
--ω h G
2 2
2
where V G = V x + V y + V z .
2
2
2
Rigid Body Dynamics
Given descriptions of inertial properties, translational and angular momentum, and kinetic energy of a
rigid body, it is possible to describe the dynamics of a rigid body using the equations of motion using
Newton’s laws. The classical Euler equations are presented in this section, and these are used to show
how a bond graph formulation can be used to integrate rigid body elements into a bond graph model.

Basic Equations of Motion
p
The translational momentum of the body in Fig. 9.30 is = m , where m is the mass, and V is the
V
velocity of the mass center with three components of velocity relative to the inertial reference frame x o ,
y o , z o . In three-dimensional motion, the net force on the body is related to the rate of change of momentum
by Newton’s law, namely,
d
F = ----- p
dt

which can be expressed as (using Eq. (9.9)),

∂ p
F = ------ + Ω × p
∂t
rel
with now relative to the moving frame x a , y a , z a , and Ω is the absolute angular velocity of the rotating
p
axes.
A similar expression can be written for rate of change of the angular momentum, which is related to
applied torques T by

∂h
T = ------ + Ω × h
∂t
rel
h
where is relative to the moving frame x a , y a , z a .

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